On expressive power of regular expressions with subroutine calls and lookaround assertions

Many regular expression engines employ syntactical extensions to provide simple, expressive support for real-world needs. These features are subroutine calls, zero-width lookaround assertions, DEFINE rules, and named parenthesised expressions. A subroutine call executes a specified subpattern where the call is placed, possibly recursively. Lookaround assertions are either lookahead or lookbehind: a lookahead is a conditional within a subpattern: when it fails, the match at the current position of the whole subpattern fails, while a lookahead itself does not consume any input; a lookbehind works as a lookahead except it checks the input prior to the current position. A DEFINE rule introduces a subpattern for use by a subroutine call, while not involved in matching where the rule is placed. A named parenthesised expression can be executed by its name in addition to the parenthesis number. This paper presents a formalisation of subroutine calls, DEFINE rules, and named parenthesised expressions using the matching relation while attempting to mimic the behaviour of real-world regular expression engines. Also, we give an alternative constructive proof of equivalence of expressive power of regular expressions extended with subroutine calls and the class of context-free languages: a conversion between such expressions and context-free grammars. Finally, the question of whether regular expressions with operations lookaround assertion combined with subroutine call have greater expressive power than expressions with only subroutine call is answered positively.